All Problems

Constant Electric Field in a Vacuum

Problem 3.17

Suppose the surface charge density over a sphere of radius \(R\) depends on a polar angle \(\theta\) as \(\sigma=\sigma_{0} \cos \theta,\) where \(\sigma_{0}\) is a positive constant. Show that such a charge distribution can be represented as a result of a small relative shift of two uniformly charged balls of radius \(R\) whose charges are equal in magnitude and opposite in sign. Resorting to this representation, find the electric field strength vector inside the given sphere.

Reveal Answer
E=−1/3kσ0/ε0, where k is the unit vector of the z axis with respect to which the angle 0 is read off. Clearly, the field inside the given sphere is uniform.\mathbf{E}=-1 / 3 \mathrm{k} \sigma_{0} / \varepsilon_{0}, \text { where } \mathbf{k} \text { is the unit vector of the } z \text { axis with respect to which the angle 0 is read off. Clearly, the field inside the given sphere is uniform.}
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